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Analogy Between Linear Optical Systems and Linear Two-Port Electrical Networks University of New Orleans ScholarWorks@UNO Electrical Engineering Faculty Publications Department of Electrical Engineering 10-1972 Analogy Between Linear Optical Systems and Linear Two-Port Electrical Networks Rasheed M.A. Azzam University of New Orleans, [email protected] N. M. Bashara Follow this and additional works at: https://scholarworks.uno.edu/ee_facpubs Part of the Electrical and Electronics Commons Recommended Citation R. M. A. Azzam and N. M. Bashara, "Analogy Between Linear Optical Systems and Linear Two-Port Electrical Networks," Appl. Opt. 11, 2210-2214 (1972) http://www.opticsinfobase.org/ao/ abstract.cfm?URI=ao-11-10-2210. This Article is brought to you for free and open access by the Department of Electrical Engineering at ScholarWorks@UNO. It has been accepted for inclusion in Electrical Engineering Faculty Publications by an authorized administrator of ScholarWorks@UNO. For more information, please contact [email protected]. Analogy Between Linear Optical Systems and Linear Two-Port Electrical Networks R. M. A. Azzam and N. M. Bashara Attention is called to the analogy between linear optical systems and linear two-port electrical networks. For both, the transformation of a pair of oscillating quantities between input and output is of interest. The mapping of polarization by an optical system and of impedance (admittance) by a two-port network is described by a bilinear transformation. Therefore for each transfer property of a system of one type, there is a similar property for the system of the other type. Two-port electrical networks are synthesized whose impedance-(or admittance-) mapping properties are the same as the polarization-mapping prop- erties of a given optical system. The opposite problem of finding the optical analogs of two-port net- works is also considered. Besides unifying the methods of handling these two different kinds of systems, the analogy appears fruitful if used reciprocally to simulate electrical networks by optical systems, and vice versa. Linear mechanoacoustic systems have optical analogs besides their well-known electrical analogs. 1. Introduction outgoing waves are monochromatic and of the same The advent of microwaves and more recently of lasers frequency, and both are totally polarized. The polari- has stimulated the interest of electrical engineers in zation states of these waves can be specified with refer- optics and optical systems. Establishing analogies ence to any chosen pair of basis states, which are gener- between electrical networks and optical systems would ally elliptic. Any polarization state can be generated be useful in (1) unifying the methods of treating both by a linear superposition of the basis states 1 and 2 using kinds of systems and (2) the reciprocal simulation of complex coefficients E and E2, respectively. These systems of one type by systems of the other. coefficients transform linearly after propagation through the system S 11. Basis of the Analogy /El'>\ T., T12 (El\ (1) The basis of the analogy between optical systems and \E2'/ ( T2, T22 E2 two-port electrical networks' is explained with reference where Tij denote the complex elements of the generalized to Fig. 1. For the optical system S, shown in Fig. 1 (a), Jones matrix2 of the optical system. The matrix T nonlinear optical effects and other frequency-changing depends both on the internal structure of S and on the phenomena (such as Brillouin or Raman scattering) are basis states used at the input and output of the optical assumed absent. Also, incoherent scattering processes system, which need not be the same. When two orthog- are excluded. Under these conditions the incident and onal linear polarizations are chosen as basis states, Eq. (1) becomes W vi (2) E Y \T21 T22IEI where (T,") refers to the ordinary Cartesian Jones ma- (a) (b) trix and the elements of the vector E correspond to the Fig. 1. (a) Linear optical system S; (b) linear two-port network Cartesian components of the electric vector of the light W. For S and W the transformation of a pair of oscillating wave. quantities between input and output is of interest. In many cases we are interested in the polarization- mapping properties of the system S overlooking any over-all amplitude or phase changes between input and, The authors are with the Electrical Materials Laboratory, output. The polarization forms (or the ellipses of Department of Electrical Engineering, University of Nebraska, polarization) of the incident and outgoing waves are Lincoln, Nebraska 68508. determined by the complex ratios X and x', respectively, Received 13 March 1972. where 2210 APPLIED OPTICS / Vol. 11, No. 10 / October 1972 x = E2/E,, x = E2//Ell' (3) (?-- - f72) = ( 3 - - 2)/(l3 - 772) (3 - 1)] According to Eq. (1) x' and x are interrelated by X [(¢ - 1)/(¢- 2)1- (9) x' = (T22x + T21)/(Tl2x + T1), (4) This means that if the transformation of three polariza- tion states by an optical system S or three impedances which is a bilinear transformation. The result in Eq. by a network W is known, the mapping of all other (4) has been utilized'- 6 to investigate various aspects of polarizations or impedances is completely determined. polarization transfer by optical systems and to provide The bilinear transformation is referred to as the polari- a procedure' (generalized ellipsometry) to measure the zation transfer function (PTF) or the impedance matrix T. transfer function (ITF). According to Eqs. (4) ahd (7), The two-port network shown in Fig. 1(b) is assumed the coefficients of the PTF and the ITF determine the linear and may contain active elements operating in the matrices T and W of Eqs. (1) and (5), respectively, up to linear domain.7 The voltage and current V' and I' at a complex multiplier. Besides the mapping of three one port are related to the voltage and current V and I polarizations or three impedances, a fourth measure- at the other port by a linear transformation ment is needed to determine these matrices completely. ( Wv1 12 \ I \ This involves the determination of the over-all phase (I) = delay and attenuation of a light wave of a given polari- W21 W22 V} zation by the optical system and a complex voltage- or where Wfj are the complex elements of the network 8 current-transfer ratio for the network. transfer matrix. The impedances at the two ports For any choice of the basis states Eq. (3) leads to a are defined by complex-plane representation in which the loci of elliptic = V/I, Z = V'/I', (6) polarizations of the same azimuth and of the same ellip- ticity constitute two orthogonal sets of circles. These and according to Eq. (5) they are related by equiazimuth and equiellipticity families of circles re- Z = (W Z + W )/(Wl Z + W11), (7) spectively pass through and enclose the two points in 2 2 21 2 the complex plane that represent the right and left which is a bilinear transformation. Obviously, the circular polarizations. If the latter polarizations are transformation of admittance between the two ports is chosen as the basis states, the equiazimuth and equi- also bilinear, ellipticity contours become straight lines and concentric circles through and around the origin, respectively. In Y' = (W,1 Y + W12)/(W21Y + W22). (8) the impedance plane these straight lines and circles The analogy between optical systems and two-port through and around the origin correspond to the loci of networks is clearly demonstrated by the similarity of impedances of constant angle and of constant magnitude, Eqs. (1) and (5) and Eqs. (4) and (7) or (8). In both respectively. The origin and the point at infinity cor- cases we deal with the transformation of (the complex respond to the left and right circular polarizations in one amplitudes of) a pair of oscillating quantities between case and to zero impedance (short circuit) and infinite the input and output. These are the two oscillating impedance (open circuit) in the other. components of the light vector along the two chosen An important property of bilinear transformations basis states in the case of optical systems and the voltage between two complex planes is that a circle in one plane and current oscillations in the case of two-port networks. is mapped into a circle in the other plane. Applying Rumsey has previously shown the similarity of the this property, it is seen that if an incident polarization definition of impedance and polarization ratios and x is of a fixed azimuth (ellipticity), the polarization x' suggested the use of impedance charts (Smith or Carter) exiting from the optical system S will traverse a -circle 9 to represent elliptical polarizations. Deschamps briefly in the complex plane as the ellipticity (azimuth) of x is mentioned the similarity between the mapping of varied to scan all possible values.' Similarly, for the polarization by optical systems and of impedance by two-port network, W, if an impedance Z of constant 0 electrical networks. ' angle (magnitude) is connected at one port of the inpe- 111. Analogous Properties of Optical Systems dance, Z' that appears at the other port will follow a circle in the complex plane as the magnitude (angle) of and Two-Port Networks Z is varied to cover all possible values. The optical From the previous section we have seen that the system S and the two-port network W map the two mapping of polarization by an optical system and of orthogonal families of straight lines and concentric impedance (or admittance) by a two-port electrical circles through and around the origin in the polariza- network between input and output is described by a tion-impedance (-Z) plane into two orthogonal sets of bilinear transformation. It follows that for each ter- circles in the ('-Z') plane (Fig.
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